which is analogous to the distance which we require to determine. We shall now give the investigation of Helmholtz, by which the analytical form of the function expressing the distance is to be ascertained (Gottingen Nachrichten, 1868, pp. 193 sq.). It must be remembered that our definition of a point will be purely analytical. Suppose three different scales of pure quantity each extending from -co to + GO . Each of these scales is perfectly continuous, so that, no matter how close any two elements in the scale may be, it is always possible to conceive the insertion of an infinite number of intermediate elements. A point is to be defined for our present purpose as a group of three numerical magnitudes taken one from each of the three scales. This conception may be stated more generally. We can conceive n different numerical scales. Then a grottp of n numbers, one from each scale, will define an element of a continuous re-fold manifoldness. It will be obvious that unless the theory of distance possess a special character it will not be possible for a rigid body to exist. Take, for instance, five points in a rigid body ABODE. There are ten different pairs of points and ten corresponding distances ; all these ten distances must remain unchanged when the body is displaced. We may assume the position of A arbitrarily. Then after the dislpacement B must be placed at the right distance from A, but will only be limited by this condition to a certain surface, C must be placed at the right distance from A and from B, thus C will be limited to a certain curve, D must be placed at the proper distances from A, from B, and from C. These conditions will be sufficient to define D with complete definiteness. In the same way E will be completely defined by its distances from A, B, and C, but as D and E are thus fully defined we have no guarantee that the distance DE shall retain, after the translation, the same value which it had before. This then indicates that the function which is to express the distance must have a very special form. Any arbitrary function of the six coordinates of the two points would in general not fulfil the condition that the distance DE after the transforma tion will retain the same value as it had before. If a greater number of points than five be taken, the conditions which a rigid system must fulfil become still more nume rous. Let x, y, z be the coordinates of a point in a rigid body free to rotate around a point. We shall assume that x, y, z is in the vicinity of the fixed point, and that the displacement of the body is such that a second point remains unaltered. Tn other words, the displacement is to be a rotation around a line joining the two points, and ve shall also assume that when this rotation has been completed every point will be restored to its original position. Let i be the angle of rotation around the axis, then x, y, z will all be functions of vj, and we may assume that the following equations will hold dr] In the first place it is plain that these differential coefficients must be functions of x, y,z, and, these functions being expanded in ascend ing powers, we may omit all powers above the first. It will also be obvious that the absolute terms must be zero as the origin is by hypothesis to be a fixed point. As the displacement is a rotation, it follows that the differential coefficients must be zero for certain values of x, y, z different from zero, but this involves the condition = 0. We now proceed to solve the three linear differential equations by the well-known process. If wo multiply the three equations by I, m, n respectively, and if we determine I, m, n so that Ih = Ia + ma-^ + na. + nc. 2 , where h is another constant determined by the equation a h , KI , # a = , b , b c , c- the differential equations then give - (Ix + my + nz) = h(lx + my + nz) , whence Ix + my + nz = Cc hr i . We have already seen that one of the values of h must be zero, whence if the other values be h l and h. 2 we have the three equations l. 2 x + m> 2 y + n, 2 z = Cc 7 ^ . It is plain that h^ and A 2 cannot be real quantities, for then the quantities l^x + 111$ + n-^z and l. 2 x + m<$ + n z z could attain any values from- oo to + co according to the variations in rj. If h l and h. 2 are imaginary then will also the corresponding values of I, m, n be imaginary. We therefore write 7^ = +<mi h. 2 = - coi L == A0 T A^ i so that A a; + ^y + v z = Ae 6r i cos (co?/ + c) tX + ^y + v^ = Ae 6r > sin (<ay + c) ; in which case we have But it is plain that unless 9 be zero the left-hand side of tliis equation will be susceptible of indefinite increase, which is contrary to our hypothesis. We are therefore entitled to assume that = 0. The two roots of the cubic for h must, therefore, be pure imaginaries, and thus we have the condition ffl + &! + c 2 = . Finally we have for the determination of x, y, z the following threo equations : I x + m y + n z = const. z = A cos (wy + c) = A sin (cay + c) . It will simplify the subsequent calculations if we now make such a transformation of the coordinates as will enable us to write We shall then have from the results just obtained = dZ -j- = +o> . dt The movement corresponding to t is such as leaves unaltered all points of which the Y and the Z are equal to zero. 1 Let us now suppose another displacement to be given to the system by a rotation 77 about another axis so chosen that all the points for which X = and Z = shall remain unaltered. For this condition to be fulfilled we must have the equation for tlien each side of these equations will be equal to zero for points which make X and Z zero. The condition that the roots of h shall
be purely imaginary gives us